Maxwell s Equations:

Transcription

1 Course Instructor Dr. Raymond C. Rumpf Office: A-337 Phone: (915) Maxwell s Equations: Terms & Definitions EE-3321 Electromagnetic Field Theory Outline Maxwell s Equations Terms and Definitions Types of Electric Current Duality Maxwell's Equations -- Terms & Definitions lide 2 1

2 Maxwell s Equations Forms of Maxwell s Equations Time-Domain Dds dv V Bds 0 Integral Form v L L Most general form B E d ds t D H d J ds t Differential Form D v B 0 B E t D H J t Frequency-Domain Dds dv V Bds 0 v L L E d jb ds H d J jd ds D v B 0 Most common form E jb H J jd Constitutive Relations: D E B H Maxwell's Equations -- Terms & Definitions lide 4 2

3 Notes About Maxwell s Equations The time-domain integral form is the most general. The frequency-domain differential form is the most common. Maxwell s equations are rarely directly useful. We derive useful equations from them. Maxwell s equations do not directly describe how electromagnetic fields interact with materials. This information comes from the constitutive relations. Electric fields do not directly experience permeability. Magnetic fields do not directly experience permittivity. Maxwell's Equations -- Terms & Definitions lide 5 Terms and Definitions 3

4 Two Forms of Electric Energy Electric energy can exist in at least two different forms: E E and P P The mere fact that we can see stars and transmit radio waves through space means that electric energy can exist in a vacuum. We know from capacitors that electric energy can be stored inside of a material when the material becomes polarized. Maxwell's Equations -- Terms & Definitions lide 7 Electric Field Intensity, E (V/m) Think of the electric field intensity as the pure form of electric energy that exists in the vacuum of space. The electric field intensity E is most closely related to voltage V and force F. It is the initial push on a dielectric medium. E 0 E Force on a Charge, F F QE Protons and electrons have opposite charge so they are forced in opposite directions in the presence of an applied electric field. Maxwell's Equations -- Terms & Definitions lide 8 4

5 Electric Polarization, P (C/m 2 ) When an electric field is applied to a material, it puts a mechanical force on all charged particles. Electrons and protons have opposite charge so they are pushed in opposite directions. Free charges cause a conduction current. Bound charges create electric dipoles that induce their own electric fields. A material said to be electrically polarized when these dipoles are present. E 0 E P Electric Polarization, P P E 0 e The electric susceptibility e is a measure of how easily a material can be polarized due to an applied electric field. Maxwell's Equations -- Terms & Definitions lide 9 Electric Flux Density, D (C/m 2 ) The electric field induced by dipoles within a material combines with external electric fields to produce an overall electric flux. The combined electric flux is the electric flux density. Note: 0 is the free space permittivity and multiples E so that 0 E has the same units as P. D 0E P Vacuum response Material response The electric flux density is most closely related to charge. Maxwell's Equations -- Terms & Definitions lide 10 5

6 Two Forms of Magnetic Energy Magnetic energy can exist in at least two different forms: H H and M M The mere fact that we can see stars and transmit radio waves through space means that magnetic energy can exist in a vacuum. We know from inductors that magnetic energy can be stored inside of a material when the material becomes magnetized. Maxwell's Equations -- Terms & Definitions lide 11 Magnetic Field Intensity, H (A/m) Think of the magnetic field intensity as the form of magnetic energy that exists in the vacuum of space. The magnetic field intensity is most closely related to current. It is the initial torque put on a magnetic dipole. H Magnetic dipoles naturally occur due to circulating charge at the atomic scale. An external magnetic field will act to align the magnetic dipoles. Maxwell's Equations -- Terms & Definitions lide 12 6

7 M Magnetic Polarization, (T or Wb/m 2 ) When a magnetic field is applied to a material, it puts a mechanical torque on all magnetic dipoles. The dipoles tend to align with the external magnetic field. The magnetic dipoles induce their own magnetic fields. A material said to be magnetized when these dipoles are aligned. H M Magnetic Polarization, M M H 0 m The magnetic susceptibility m is a measure of how easily a magnetic dipole is tilted due to an applied magnetic field. Maxwell's Equations -- Terms & Definitions lide 13 Magnetic Flux Density, (T or Wb/m 2 ) The magnetic field induced by dipoles within a material combine with external magnetic field to produce an overall magnetic flux. The combined magnetic flux is the magnetic flux density B. Note: 0 is the free space permeability and multiples H so that 0 H has the same units as M. B B 0H M Vacuum response Material response The magnetic flux density is most closely related to force. Compass needles align in the direction of the B field. Maxwell's Equations -- Terms & Definitions lide 14 7

8 Electric Current Density, J (A/m 2 ) The electric current density is the electric current per unit area. At low frequencies, current utilizes the entire volume of a conductor. The current is distributed over the entire cross section of a conductor. At high frequencies, the skin effect must be considered. I total enclosed current I J ds Maxwell's Equations -- Terms & Definitions lide 15 Volume Charge Density, v (C/m 3 ) The volume charge density is the electric charge per unit volume. Charge tends to distribute itself through a volume. V v Q total enclosed charge Q V dv v Maxwell's Equations -- Terms & Definitions lide 16 8

9 Permittivity, (F/m) The permittivity is a measure of how well a material stores electric energy. It can be thought of as a measure of how much interaction an electric field has with a material. Permittivity is most closely related to capacitance. r 0 r F m 0 1 no units permittivity 0 vacuum permittivity relative permittivity dielectric constant r Maxwell's Equations -- Terms & Definitions lide 17 Permeability, (H/m) The permeability is a measure of how well a material stores magnetic energy. It can be thought of as a measure of how much interaction a magnetic field has with a material. Permeability is most closely related to inductance. r 0 r H m 0 1 no units permeability 0 vacuum permeability relative permeability r Maxwell's Equations -- Terms & Definitions lide 18 9

10 Conductivity, (1/m) The conductivity describes the degree to which a material conducts electricity. positive charge negative charge I J E Resistivity Ωm 1 Ohm s Law for EM J E Total Current I J Maxwell's Equations -- Terms & Definitions lide 19 Types of Electric Current 10

11 Types of Electrical Current Conduction Current, Flow of free charges through conductors. Traditional kind of current studied in circuit theory. Convection Current, Flow of free charges through insulators. Displacement Current, Momentary movement of bound charges in a dielectric. Total Current: T J J J D J J J J D Maxwell's Equations -- Terms & Definitions lide 21 Conduction Current, J Conduction current is the flow of free charges through a conductor. Ohm s law for electromagnetics J E Maxwell's Equations -- Terms & Definitions lide 22 11

12 Convection Current, J Convection current is the flow of free charges through an insulator. This current does not obey Ohm s law since no conductors are involved. For this to exist, the insulators must be charged. Convection Current J v v Continuity of Current Equation v J t Maxwell's Equations -- Terms & Definitions lide 23 Displacement Current, J D There exists short-lived net movement of charge as a material changes its state of polarization. J D Displacement Current J D D t Maxwell's Equations -- Terms & Definitions lide 24 12

13 Total Current, J T The total current is the sum of conduction current, convection current, and displacement current. J J J J T Current due to free charges J This is the J term in Maxwell s equations. It describes only the movement of free charges. D Current due to bound charges D t Maxwell's Equations -- Terms & Definitions lide 25 Duality 13

14 ymmetric Form of Maxwell s Equations To illustrate and to utilize duality, two non-physical terms are added to Maxwell s equations. These are the magnetic charge density m and the magnetic current density J m. D e B m D E B H B E J m t D H Je t Aside: The nonphysical terms have applications in analysis. For example, interfaces with high dielectric contrast can sometimes be approximated as a perfect magnetic conductor. Maxwell's Equations -- Terms & Definitions lide 27 Duality of the Parameters We are now in a position to summarize the duality of the parameters. D e B m B E J m t D H Je t Electric Quantity E (V/m), electric field intensity D (C/m 2 ), electric flux density J e (A/m 2 ), electric current density e (C/m 3 ), electric charge density (F/m), permittivity Magnetic Quantity H (A/m), magnetic field intensity B (Wb/m 2 ), magnetic flux density J m (Wb/sm 2 ), magnetic current density m (Wb/m 3 ), magnetic charge density (H/m), permeability Maxwell's Equations -- Terms & Definitions lide 28 14

15 Notes on Duality elf-check for Derivations -- We can derive analogous equations for electric and magnetic quantities. Aside from a sign, the equations should be the same after a change of variables. This is a great self check! Design Freedom -- We can swap and and the device will behave the same, but with E and H swapped. Maxwell's Equations -- Terms & Definitions lide 29 15